Connecting electronic entropy to empirically accessible electronic properties in high temperature systems

Connecting electronic entropy to empirically accessible

electronic properties in high temperature systems

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.

Citation Rinzler, Charles C., and Antoine Allanore. “Connecting Electronic Entropy to Empirically Accessible Electronic Properties in High Temperature Systems.” Philosophical Magazine 96, 29 (September 2016): 3041–3053 © 2016 Informa UK limited, trading as Taylor & Francis group

As Published https://doi.org/10.1080/14786435.2016.1216657

Publisher Taylor & Francis

Version Author's final manuscript

Citable link http://hdl.handle.net/1721.1/114780

Terms of Use Creative Commons Attribution-Noncommercial-Share Alike

Connecting Electronic Entropy to Empirically Accessible Electronic Properties

in High Temperature Systems

Charles C. Rinzler

_{and Antoine Allanore}

1_{Massachusetts Institute of Technology, Department of Materials Science and Engineering} 77 Massachusetts Avenue, Room 13-5066, Cambridge, MA, USA 02139

e-mail address: allanore@mit.edu

617-452-2758

2_{Massachusetts Institute of Technology, Department of Materials Science and Engineering} 77 Massachusetts Avenue, Room 13-5095, Cambridge, MA, USA 02139

e-mail address: rinzler@mit.edu

Connecting Electronic Entropy to Empirically Accessible Electronic Properties

in High Temperature Systems

A quantitative theoretical model connecting the thermopower and electronic entropy of molten systems is proposed, the validity of which is tested for semiconductors and metallic materials. The model accurately provides the entropy of mixing for molten semiconductors, as shown for the representative system Te-Tl. Predictions of the electronic entropy of fusion for compounds are in agreement with available data and offer a novel means to identify the correct electrical conductivity model when Hall measurements are not available. Electronic entropy for molten semiconductor and metallic systems is shown to reflect order in the molten and solid state. The model proves accurate at predicting the electronic state entropy contribution to the electronic entropy of mixing.

Keywords: entropy; electronic entropy; thermopower; molten semiconductor

PACS: 64, 71, 81

1. Introduction

Electronic entropy is a manifestation of the accessible state-space of electrons and consequently reflects the structure of condensed matter systems. Empirical access to electronic entropy therefore offers a means of probing the electronic ordering of matter. To date, in materials

thermodynamics, the contribution of electronic entropy to the total entropy or, more importantly, to the entropy variation across phases, is often ignored or neglected [1]. As claimed in Fultz [2], the study of electronic entropy remains a frontier for high-temperature thermodynamics. A

at high temperature. The model is first used to successfully predict the thermopower and entropy of mixing of the molten Te-Tl system with accuracy comparable to other empirical methods. The robustness of the model is then tested for the electronic entropies of fusion of selected compounds and for the entropy of mixing of a molten metallic system. The proposed approach proves to offer novel access to ordering and mixing properties in molten metallic and semiconductor systems. We show that in certain cases, here at high temperatures, electronic entropy contributes significantly to the total entropy of a system.

2. Electronic Entropy and Thermopower

We propose to measure the total electronic entropy of high temperature systems through the measurement of macroscopic electronic properties. The microscopic basis for the connection between thermopower and entropy has previously been discussed by Peterson and Chaikin [3,4], and the statistical mechanical basis for the electronic entropy has been presented in Wallace [5]. Prior work by Rockwood has proposed the following connection between the partial molar electronic entropy of a system (𝑆̅ , a macroscopic thermodynamic property of the system, in J _𝑒 mole--1 K-1)and the thermopower [6–8]:

𝑆̅ = −𝐹𝛼, _𝑒 (1)

where we understand 𝑆̅ as defined as: _𝑒

𝑆̅ = (_𝑒 𝑑𝑆

where 𝑆 is the total entropy (J mol-1 _K-1_{), 𝑛}

𝑒 refers to the number of moles of mobile electrons per

mole of system, 𝐹 is Faraday’s Constant,and the thermopower (𝛼, VK-1_{) is classically defined in}

terms of a measured difference in electrical potential (𝜙) as a function of temperature [9]:

∆𝜙

∆𝑇 = −𝛼. (3)

The reader is invited to consider Rockwood, Kjelstrup, and Tykodi for a historical

perspective on the connection between thermopower and electronic entropy [8,10–12]. It should be noted that there has been much discussion in the literature, starting with Thompson, about the possibility of a quantitative connection between irreversible thermodynamic properties, such as the thermopower or Peltier coefficient, and reversible thermodynamic properties, such as the entropy or heat capacity.

We seek to provide empirical evidence that quantitatively connects thermopower to electronic entropy. To leverage available experimental entropy data, it is useful to work in the context of integral instead of partial molar entropy. From Eq. (2) we can define:

𝑆_𝑒𝑝 = 𝑛𝑒𝑆̅ . 𝑒 (4)

Leveraging Eq. (1) we can reframe Eq. (4) in terms of the thermopower:

𝑆_𝑒𝑝 = −𝑛_𝑒𝐹𝛼, ₍₅₎

in units of J mol-1 _K-1_{or, equivalently:}

in units of J m-3 _K-1 _{where 𝑛 is density of electrons (m}-3_{) that contribute to nonlocal transport}

phenomena and 𝑒 is the fundamental charge. For materials where the conducting particle has positive effective charge (e.g. holes) Eq. (6) reads:

𝑆_𝑒𝑝 = 𝑛𝑒𝛼, ₍₇₎

where 𝑛 is now the density of positive charge carriers that contribute to nonlocal transport phenomenai_.

Thus, it is in principle possible to macroscopically probe the electronic properties of a system (thermopower and number of carriers) and relate them to an essential thermodynamic quantity: entropy. As can be demonstrated starting from thermodynamics of solutions (see Appendix), electronic entropy is the critical driving factor for the thermodynamic behavior of certain systems, including molten semiconductors. In such systems, evaluating the electronic entropy of mixing, Δ𝑆_𝑒, provides an approximation of the total entropy of mixing Δ𝑆_𝑚𝑖𝑥.

3. Application to Molten Semiconductors

To verify the proposed connection both electronic and thermodynamic data of high temperature systems are required. There is a dearth of available electronic data at high temperatures.

However, one class of material system, molten semiconductors, has been studied in detail precisely because of the comprising members’ unique electronic properties. Consequently, we will leverage the available datasets for these systems to evaluate the predicting performances of the proposed model.

Molten semiconductors exhibit electronic properties characteristic of solid semiconductors in their liquid phase (the molten state). Herein the term semiconductor refers to systems that

have an optical band gap and an increase in electronic conductivity with temperature. Sulfides, oxides, selenides, tellurides, and other systems of commercial relevance exhibit these

features [13]. A strong body of work describes the physical basis of the electronic behavior of molten semiconductors [14] and indicates that for many systems the Drude model holds for electronic conductivity [15].

Applying the Drude model of conductivity to mobile electrons allows connecting the density of mobile electrons, 𝑛, to two empirically accessible quantities, the conductivity (𝜎, ohm-1

m-1_{) and mobility (𝜇, m}2 _V-1 _s-1_):

𝜎 = 𝑛𝑒𝜇. (8)

For materials for which the Drude model holds, the connection between 𝑆_𝑒𝑝 _{and the}

electronic properties therefore follows for negatively charged carriers:

𝑆_𝑒𝑝 = −𝜎

𝜇𝛼, (9)

and for positively charged carriers:

𝑆_𝑒𝑝 =𝜎

𝜇𝛼. (10)

4. Method to Predict Thermodynamic and Electronic Properties

We therefore propose a method to quantitatively connect the theoretically and experimentally well-established electronic properties of molten semiconductor systems and the entropy of mixing. The following assumptions underlie the method, informed by the above discussions on the

(1) The entropy of mixing is accounted for by the electronic entropy of mixing

(2) The Drude model of conductivity describes the conductivity of the system

(3) The Hall mobility is used for the carrier mobility

(4) The thermopower is dominated by the electronic contribution

(5) The total electronic entropy is approximated by 𝑆_𝑒𝑝

Assumption (5) is read:

𝑆_𝑒 = 𝑆_𝑒𝑝 ₍₁₁₎

where 𝑆_𝑒 is the total electronic entropyii _[1].

By means of example the method is tested to predict the thermopower of the tellurium (Te)-thallium (Tl) system at 800°K. As one of the most studied molten semiconductor systems, both thermodynamic and electronic data are available for the tellurium-rich side of the phase diagram and for pure tellurium and thallium. The Hall mobility, thermopower, and electronic conductivity at 800°K are provided by Cutler as a function of composition [16–18].

The thermopower is connected to the total electronic entropy, according to Eqs. (9-11). The electronic entropy mechanical mixingiii _{term 𝑆}

𝑒𝑚 is obtained from the electronic entropy of the

end-members:

𝑆𝑒𝑚 = 𝑥𝑇𝑙𝑆𝑒𝑇𝑙+ 𝑥𝑇𝑒𝑆𝑒𝑇𝑒. (12)

𝑆_𝑒𝑇𝑙 _{and 𝑆}

𝑒𝑇𝑒 are absolute electronic entropies obtained via Eqs. (6), (7), and (11) from the

reported electronic properties of the end members in the molten state. We note that other methods are available to obtain the total electronic entropy of pure substances such as metals (i.e.

ab initio atomistic modeling) [2]. The total electronic entropy is then obtained from the electronic entropy of mixing and mechanical mixing terms:

𝑆𝑒 = ∆𝑆𝑒+ 𝑆𝑒𝑚. (13)

Substituting 𝑆𝑒 in Eq. (13) with Eqs. (9-11) and using the assumption that ∆𝑆𝑒 ≈ ∆𝑆𝑚𝑖𝑥 (as

discussed in the Appendix), the thermopower is obtained:

𝛼 =𝜇

𝜎(∆𝑆𝑚𝑖𝑥+ 𝑥𝑇𝑒𝑛𝑇𝑒𝑒𝛼𝑇𝑒+ 𝑥𝑇𝑙𝑛𝑇𝑙𝑒𝛼𝑇𝑙). (14)

5. Results for Tl-Te System

We restrict our calculation to the tellurium-rich portion of the phase diagram, the only

compositions for which Hall mobility measurements are available for the melt. The entropy of mixing (∆𝑆_𝑚𝑖𝑥) is provided by Nakamura and Terpilowski from electromotive force (e.m.f.) measurements [19,20]. Figure 1a shows the predicted thermopower of Te-Tl from 0 to 67% atomic fraction of thallium (the congruent melting composition) by Eq. (14), compared to empirical data provided by Cutler.

Reversing the connection, we can demonstrate the utility of electronic property

measurements of the molten state as a means to quantify entropy in the molten phase. We use reported empirical measurements of the mobility, electronic conductivity, and thermopower to calculate the entropy of mixing by Eq. (14). The results are shown in Fig. 1b against empirical data from Nakamura and Terpilowski and a model of the entropy of mixing developed by Oh based on Hillert’s two-sublattice model and optimized with the data from Nakamura and Terpilowski.

Figure 1b shows that experimental uncertainty for e.m.f. measurements of the entropy of mixing for this system is on the order of 1 Jmol-1 _K-1_{. The modeled entropy of mixing by Oh is fit to}

this experimental dataset. The predicted entropy of mixing by Eq. (14) (the solid line in Fig. 1b) shows maximum deviation from the modeled data of less than 0.3 Jmol-1 _K-1_{, well within the range}

of experimental uncertainty of the e.m.f. method and without the use of any fitting procedure. Errors in the measurement of the electronic properties propagate through the model to errors in the predicted entropy of mixing. Specifically, a sensitivity analysis of error propagation for measurement of the thermopower shows that a 10% error in the thermopower propagates to roughly a 10% error in the entropy of mixing.

Measurements of thermopower in high temperature molten systems are typically reported with errors of less than 10% [21]. The conductivity is typically reported with measurement error of less than 1.5% [22]. Consequently, for systems such as Te-Tl, with entropies of mixing on the scale of < 5 Jmol-1 _K-1 _{that are primarily comprised of electronic entropy, the use of the present}

model using traditional experimental methods is expected to result in errors of < 0.5 J mol-1 _K-1_.

6. Discussion on the Validity of the Model

We made the assumption that the total electronic entropy is approximated by 𝑆_𝑒𝑝 _{(Eq. (11)).}

This, as shown by Rockwood for the case of a free electron system, is not always an appropriate assumption [7]. However, as shown in Figures 1a and 1b, our assumption led to the accurate prediction of macroscopic properties of a molten semiconductor system. We interpret the results to indicate the validity of assumption (5) for molten semiconductor systems.

Further, as shown in Figures 1a and 1b, the predicted total electronic entropy of mixing approximates the total entropy of mixing for the Te-Tl system. This validates assumption (1), in

accordance with our expectations based on macroscopic thermodynamic considerations (Appendix): the electronic entropy is critical to the thermodynamics of mixing for molten semiconductors.

The total electronic entropy is related to the accessible density of states (DOS) of electrons; only states vicinal to the chemical potential of electrons (or the Fermi level) are accessible and substantially contribute to the electronic entropy of the system.

Measuring the thermopower of a system therefore consists of measuring the response of the electronic component of the entropy to a perturbation in temperature. This measurement is not without analogy to electromotive force (e.m.f.) measurement, one of the few experimental methods to determine the (total) entropy of a system by monitoring a relative chemical potential difference as a function of temperature [23].

If the thermopower of a system is quantifiably related to the electronic entropy and gives access to the absolute electronic entropy of a system then the thermopower can be considered a material property with physical meaning for a material at equilibrium [23,24].

If Eq. (6) is valid for semiconductor systems, then we may draw an interesting analogy between the entropy of transport used in the irreversible thermodynamics derivation of the thermopower and the electronic entropy, a reversible thermodynamic quantity. In irreversible thermodynamics, the thermopower reflects the entropy transported during the thermally driven motion of charged particles. The thermopower is then defined as the entropy per unit charge associated with mobile electrons in the system [25].

Therefore, the thermopower of a material and the number of charged particles that

contribute to nonlocal transport phenomena (i.e. mobile electron density 𝑛 of charge 𝑒) provide a quantification of the total transported entropy of a system (𝑆_𝑡):

𝑆𝑡 = −𝑛𝑒𝛼 (15)

in units of J m-3 _K-1 _{which is identical to Eq. (6). From the equivalence, we can conclude that, for}

molten semiconductor systems, the irreversibly defined transport entropy is equal to the reversibly defined total electronic entropy:

𝑆𝑡 = 𝑆𝑒. (16)

Rockwood discusses the implications of the connection between irreversible and reversible thermodynamic quantities in [10].

7. Extension to Solids and Other Semiconductor Systems

It is foreseen that the connection of the electronic properties to electronic entropy will provide novel insights even if some of the assumptions of the method do not hold, such as the total entropy of mixing being approximated by the electronic entropy of mixing, the Drude model

approximating the conductivity of the system, or the validity of Eq. (11). To illustrate the value of the connection, it is applied to predict the entropy of fusion, and in particular the electronic entropy of fusion ∆𝑆𝑓 of solid semiconductor compounds - a property for which there exist more

data at present. To apply the proposed model, we use available data on the electronic properties of the solid and molten compound at melting and apply Eqs. (9) and (10). We focus on binary

systems for which these data, as well as electronic entropy of fusion data, are available in the literature.

Figure 2 plots the predicted electronic entropy of fusion via Eqs. (9) and (10) vs. the electronic entropy of fusion as provided by Belotskii [26].

Figure 2 shows that for certain systems Eqs. (9) and (10)provide a reasonable prediction of the reported value of the electronic entropy of fusion (solid symbols). However, the prediction can overestimate and underestimate the reported value of the entropy of fusion (hollow circles). Addressing these errors in turn helps to illustrate the value of the correlation.

The two compounds for which Eqs. (9) and (10)substantially overestimate the electronic entropy of fusion are AlSb and GaSb. Hall measurements of AlSb and GaSb report electron

densities of 7.0 x 1021 _cm-3 _{and 4.1 x 10}22 _cm-3 _{respectively, as opposed to 2.0 x 10}23 _cm-3 _{and 2.2 x}

1023 _cm-3 _{predicted by the Drude model of conductivity (Eq. (8)) [27]. Adopting the measured}

electron density of GaSb and applying Eqs. (6) and (7) predicts an electronic entropy of fusion of 14.5 Jmol-1 _K-1 _{vs. the literature provided value of 16.7 Jmol}-1 _K-1 _{– a prediction within the}

anticipated error of the reported values [26]. The corrected value is shown in Fig. 2 (solid blue symbol). Not only does this demonstrate the validity of the proposed connection for systems whereby the Drude model is inapplicable, it demonstrates the utility of the connection to determine which electronic model is best suited for systems where thermodynamic data are available. This is particularly valuable since Hall measurements for molten systems are not as common as for their solid counterparts. We have compiled these data and provided corrected predictions via Eqs. (6) and (7) in Fig. 2 as available (blue symbols).

The model does not accurately account for the entirety of the electronic entropy of fusion for some compounds. We put forth three explanations for this. The first, described above, is the

Fig. 2, where a correction to the Drude model is applied, a systemic under-prediction of electronic entropy remains. A second explanation is the lack of validity of our assumption reflected in Eq. (11) (i.e. there are contributions to the total electronic entropy that are not accounted for by 𝑆_𝑒𝑝_).

A third explanation for the remaining discrepancy lies in the existence of a second contribution to the electronic entropy, not accounted for by the connection put forth. As indicated in Section 2, the electronic entropy described in the correlation stems from the size of the available state space of the electrons near the Fermi level. It does not take into account the configurational electronic entropy, since it does not manifest in the thermoelectric power. Configurational electronic entropy describes the configurational entropy of localized electrons and holes.

In analogy to the configurational entropy of a solid solution on a lattice, the configurational electronic entropy is described by Zhou et. al. [28]. The configurational electronic entropy has been shown to substantially contribute to the electronic entropy and the total entropy of mixing for certain classes of systems, such as mixed valence oxides [20] or certain semiconductor systems at melting [29]. Germanium is for example modeled to have a configurational electronic entropy of fusion of approximately 22.1 Jmol-1 _K-1_{, an amount that bridges the electronic entropy of fusion}

predicted by our model and the reported value. The combined predicted electronic entropy of fusion is shown in Fig. 2 (red symbol).

Whereas the configurational contribution to total electronic entropy stems from electrons that are localized, the electronic entropy described by Eqs. (6) and (7) describes the entropy of delocalized electrons. We shall henceforth call this quantity the electronic state entropy. This terminology arises from the number of available states, or density of states (DOS), of a system near the Fermi level (as accessible according to the Fermi-Dirac distribution). Thus, the

configurational electronic entropy reflects statistics of distinguishable particles whereas the electronic state entropy reflects the statistics of indistinguishable particles. For systems such as alkali metals, where the DOS is often small near the Fermi level, or for ionic systems, where the DOS is often zero near the Fermi level, the contribution of the electronic state entropy to the total electronic entropy is small. However, for systems such as semiconductors (or the subclass of thermoelectrics that are designed to have maximal density of states near the Fermi level), there can be a substantial contribution of the electronic state entropy due to the large DOS near the Fermi level. Thus, it is for molten semiconductors that Eq. (14) produces the best approximation for the total entropy of mixing.

Hence, the provided model reflects the electronic state entropy as it manifests in the DOS near the Fermi level, and to date does not include a configurational component arising from localization of electrons or holes.

8. Extension to Metallic Systems

The DOS directly reflects the chemical ordering (bonding) of a system [30]. Consequently, it would be surprising if an empirically accessible quantity that reflects the DOS near the Fermi level (i.e. the thermopower) did not give at least qualitative access to the ordering tendencies of a system. We have demonstrated above that, in the case of molten semiconductors, the electronic entropy dominates the entropy of mixing and hence substantially reflects the ordering of a system. However, even for cases where the electronic entropy of mixing is not a dominant component of the total entropy of mixing, the electronic entropy will still reflect an evolution in the chemical ordering of a system.

Figure 3 shows the predicted electronic contribution to the entropy of mixing (black circles) in the molten state according to Eq. (14) of the Mg-Bi system at 1100°K and the

empirically validated total entropy of mixing (black line) [31]. The electronic property data is provided by Ratti and Enderby [13,32]. The Mg-Bi system is metallic across the entire range of composition in the molten phase.

The total entropy of mixing for this system is on the scale of 6 J mol-1 _K-1_{, confirming that}

the electronic entropy does not describe the entire thermodynamics of mixing of the Mg-Bi system. However, the electronic state entropy of mixing suggests ordering at a composition of 40% Bi. The phase diagram of the Mg-Bi system indeed exhibits a congruent melting compound at Mg3Bi2 [33].

Thus, the electronic entropy, as analyzed through the proposed correlation and empirically probed by measurement of the electronic properties of the melt, also reflects the chemical

ordering of metallic systems. When measured near the liquidus, these properties are a proxy for ordering of the corresponding solid phases. It is therefore put forth that measurement of

electronic properties of melts can provide fast and simple access to high-temperature solid phase ordering.

9. Extension to Ionic Systems

The above connection between the measured thermopower and the electronic entropy of a system assumes that the electronic contribution to the thermopower dominates. However, in some

systems (especially ionic systems), the Soret effect substantially contributes to the measured thermopower. The Soret effect reflects the chemical response of the system to a perturbation of temperature. For such a situation, Carl Wagner has provided a comprehensive method and

analysis to isolate the electronic and ionic contribution to the thermopower [23]. We note that as electronic thermopower reflects a change in the contribution of (delocalized) electron entropy to the free energy as a function of temperature, the Soret effect reflects the contribution of chemical species entropy to the free energy as a function of temperature. We propose that, by analogy, the partial entropy of species of systems exhibiting the Soret effect should be accessible via

measurement of the thermopower. Rockwood and Tykodi discuss further implications of the ability to measure electronic entropy on the ability to quantify reversible thermodynamic quantities of ionic melts that were previously inaccessible [7,11].

10. Conclusion

We have proposed a quantitative connection between thermodynamic properties (entropy) and electronic properties (thermopower) of systems. The model quantitatively predicts the entropy of mixing of molten semiconductor systems. Electronic entropy is shown to reflect ordering in

molten systems and the solid phase, even for systems in which the electronic entropy of mixing does not substantially contribute to the total entropy of mixing. The model has been shown to have consequences in the interpretation of the connection between quantities from irreversible and reversible thermodynamics. It is proposed that an analogous method using the Soret effect

Table 1 summarizes the importance of electronic entropy for different classes of systems, and the materials information that can be derived from its experimental measurement by the method put forth in this work.

Acknowledgements

This work was supported by the AFOSR under Grant FA9550-15-1-0046. We would like to

acknowledge the role of the Fannie and John Hertz Foundation for support of the researchers and Dr. Ali Sayir and the AFOSR for their financial support.

Endnotes

i _{For systems that exhibit ambipolar conductivity a similar equation with contributions from both}

electrons and holes is used. We simplify for purpose of demonstration to systems that exhibit monopolar conductivity.

ii _{For a discussion on the validity of decomposing the free energy function into components see}

Smith [34]

iii _{Mechanical mixing entropy is the concentration-weighted entropies of the end-members of a}

system in absence of any chemical interactions. Thermodynamic mixing functions are referenced to mechanical mixing: ∆𝑋_𝑚𝑖𝑥 = 𝑋 − 𝑋𝑚.

References

[1] B. Fultz, Prog. Mater. Sci. 55, 247 (2010).

[2] B. Fultz, Phase Transitions in Materials (Cambridge University Press, 2014).

[3] M. R. Peterson and B. S. Shastry, arXiv (2010).

[4] P. M. Chaikin and G. Beni, Phys. Rev. 13, (1976).

[5] D. C. Wallace, Statistical Physics of Crystals and Liquids (World Scientific, 2002).

[6] A. Rockwood, Phys. Rev. A 30, 2843 (1984).

[7] A. Rockwood, Electrochim. Acta 112, 706 (2013).

[8] A. Rockwood, J. Mod. Phys. 199 (2016).

[9] R. B. Roberts, Philos. Mag. 36, 91 (1977).

[10] A. Rockwood, Electrochim. Acta 107, 686 (2013).

[11] R. Tykodi, Thermodynamics of Systems in Nonequilibrium States (Thinkers’ Press, 2002).

[12] S. Kjelstrup, Electrochem. Acta 107, 693 (2013).

[13] J. E. Enderby and A. C. Barnes, Rep. Prog. Phys. 53, 85 (1990).

[14] A. F. Ioffe and A. R. Regel, Progress in Semiconductors, v. 4 (1960).

[15] N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, 1971).

[16] J. Donally and M. Cutler, Phys. Rev. 176, 1003 (1968).

[19] C. Oh, J. Phase Equilibria 14, 197 (1993).

[20] Y. Nakamura and M. Shimoji, Trans. Faraday Soc. 67, 1270 (1971).

[21] S. Ohno, A. C. Barnes, and J. E. Enderby, J. Phys. Condens. Matter 6, 5335 (1994).

[22] V. P. Didoukh, Y. O. Plevachuk, and B. I. Sokolovskii, Int. J. Thermophys. 20, 343 (1999).

[23] C. Wagner, Prog. Solid State Chem. 7, 1 (1972).

[24] D. Adler, Physics of Disordered Materials (Plenum Press, New York, 1985).

[25] H. B. Callen, 333, (1948).

[26] D. P. Belotskii and O. N. Manik, J. Thermoelectr. 1, 32 (2004).

[27] A. R. Regel, V. M. Glazov, and A. A. Aivazov, Sov. Phys. Semicond. 8, 1398 (1975).

[28] F. Zhou, T. Maxisch, and G. Ceder, Phys. Rev. Lett. 97, 155704 (2006).

[29] B. K. Chakraverty, Radiat. Eff. 4, 39 (1970).

[30] C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

[31] R. Hultgren, R. Orr, P. Anderson, and K. Kelley, Selected Values of Thermodynamic Properties of Metals and Alloys (Wiley, 1963).

[32] V. Ratti and A. B. Bhatia, J. Phys. F 5, 893 (1975).

[33] A. Nayeb-Hashemi and J. B. Clark, Bull. Alloy Phase Diagrams 6, 528 (1985).

[34] P. E. Smith and W. F. Van Gunsteren, J. Phys. Chem. 13735 (1994).

Appendix

For simplicity, a binary solution of A and B that defines a molten semiconductor system is considered, though the formalism is not restricted to binary systems. For a solution in the A-B system, the Gibbs free energy of mixing (∆𝐺_𝑚𝑖𝑥) is the difference between the Gibbs free energy (𝐺) of the actual solution and the mechanical Gibbs free energy (𝐺𝑚_{) of the end-members A and B,}

defined as:

𝐺𝑚 = 𝑥_𝐴𝐺_𝐴 + 𝑥_𝐵𝐺_𝐵. (A1)

∆𝐺_𝑚𝑖𝑥 is conventionally expressed in terms of the enthalpy and entropy of mixing, respectively ∆𝐻_𝑚𝑖𝑥 and ∆𝑆_𝑚𝑖𝑥, as:

∆𝐺_𝑚𝑖𝑥 = 𝐺 − 𝐺𝑚 _{= ∆𝐻}

𝑚𝑖𝑥− 𝑇∆𝑆𝑚𝑖𝑥. (A2)

The entropy of mixing can be delineated into contributions from configurational (∆𝑆_𝑐), vibrational (∆𝑆_𝑣), electronic (∆𝑆_𝑒), and other terms (∆𝑆_𝑟):

∆𝑆_𝑚𝑖𝑥 = ∆𝑆_𝑐+ ∆𝑆_𝑣+ ∆𝑆_𝑒+ ∆𝑆_𝑟. _(A3)

∆𝑆_𝑒 is typically of substantial magnitude for molten semiconductor systems (see Table A1). The configurational entropy of mixing (∆𝑆𝑐) is determined by the degree of chemical ordering

present in a system. The less ordered a system, the larger the configurational entropy of mixing. Molten semiconductors, as presented above, are ordered systems and consequently have minimal configurational entropy of mixing. Such a statement can be substantiated by consideration of the

short-range order nearly identical to the solid compound (AB). The atoms of A and atoms of B are fully associated with each other at the AB composition, thus minimizing configurational entropy of mixing. Between the congruent melting point and the end members are typically miscibility gaps. For compositions within these miscibility gaps the system segregates into two non-mixing phases, which further reduces the configurational entropy of mixing. Consequently, ∆𝑆_𝑐 is typically of minor magnitude and the result is that configurational entropy plays a small role in the entropy of mixing for molten semiconductor systems:

|∆𝑆𝑐| ≪ |∆𝑆𝑒|. (A4)

The lack of long-range order in molten systems leads to a drastic reduction of the magnitude of the vibrational contribution to the entropy in comparison with solids [1]. ∆𝑆_𝑣 for molten systems reflects the difference between vibrational entropy of the molten system and the mechanical mixture of the two end members (referenced to the liquid state) – the absolute magnitude of which is not significant for the majority of molten semiconductor systems:

|∆𝑆𝑣| ≪ |∆𝑆𝑒|. (A5)

∆𝑆_𝑟 may comprise magnetic entropy and other system-specific entropic effects, which are ignored at the present stage.

TABLE A1. Typical entropies of mixing for molten semiconductor systems [26,27,35]

Value for Molten Semiconductors

∆𝑺_𝒄(J mol-1 _K-1_{) < 1}

∆𝑺_𝒗(J mol-1 _K-1_{) << 1}

Tables

Table 1. Role of electronic entropy for various molten material systems

Metallic Semiconducting

Ionic

Reflects ordering Yes Yes Yes

Approximates

(∆𝑺_𝒎𝒊𝒙) No Yes No

Empirically

accessible in melt Yes Yes Yes

Figure Captions

Figure 1. (a) thermopower vs. atomic fraction thallium at 800°K as predicted by Eq. (14) (solid line) and as measured by Cutler (blue circles) [17]. (b) entropy of mixing vs. atomic fraction thallium at 800°K as predicted by Eq. (14) (solid line), as measured by Terpilowski (red circles), as measured by Nakamura (blue circles), and as calculated by Oh using the two-sublattice Hillert model optimized with data from Terpilowski and Nakamura (dashed line) [19,20]

Figure 2. The predicted vs. reported electronic entropy of fusion [26]. Solid symbols show predictions that agree with reported values. Hollow symbols show predictions that substantially deviate from reported values. Black symbols show predictions using Eqs. (9) and (10). Blue symbols show predictions using Eqs. (6) and (7) and measured values for the number of carriers. The red symbol shows the predicted entropy of fusion for Ge including the configurational

electronic entropy of fusion.

Figure 3. The predicted electronic entropy of mixing of the Mg-Bi system at 1100°K by Eq. (14) (black circles) and literature reported values of the total entropy of mixing (solid line) [31]

Figure 1

a.